《大学物理简明教程》 课件 7-1 SHM and springs

SimpleHarmonicMotionPendulumMassonaspring(springoscillator)vmax0x-xEquilibriumpositionMaximumdisplacementMaximumdisplacementv=0v=0Oscillation“Harmonic”“Simple”Positionasafunctionoftimeisasinusoidalfunction“Simple”Harmonicmotion“Simple”:onlyonesine/cosinefunctionVerticalpositionx

asafunctionoftimetistheamplitudeistheperiodistheinitialphase

angledependsonthe“initialconditions”

isthefrequencyistheangularfrequencyOtherwaysofexpressing

x(t)

A

istheamplitude–maximumdisplacement

isthefrequencyistheangularfrequencySimpleHarmonicMotion

A

istheamplitude–maximumdisplacement

istheperiodistheinitialphase

angledependsonthe“initialconditions”

velocityandaccelerationrestoringforcemassonspringpendulumenergyofsimpleharmonicmotionfrequencyandperiodGeneralsolution:Whatisthepositionattimet=0?Initialconditions

If

Whatisthevelocity?

If

vx=A,v=0

Acceleration?

Ingeneral,av

Whyisamotion“simpleharmonic”

?

Newton’ssecondlaw

Thenetforceonanobjectisproportionaltoitsdisplacement,andopposetothedirectionofthedisplacement-Restoring

Forceor

F=-

kxRESTORINGFORCEF=-

kxequilibriumpositionNoforce!ForcetowardsequilibriumpositionproportionaltothedistancefromequilibriumF=-kxF=kxForcetowardsequilibriumpositionx-xHooke’slawTheextensionofaspringisproportionaltotheforceF=-

kxThemotionofthemasscanbeexpressedas

Simpleharmonicmotion

Equilibriumposition(withmassM)unstretchedspring(nomassattached)stretchedspring(withmassattached)thespringforceequalsthegravitationalforceEquilibrium:Averticalspringwithgravity11gravitationalpull1Netforceforcefromspring1susinganewequilibriumposition

sameasthehorizontalspring!PeriodofaspringharmonicoscillatorRestoringforce:Harmonicmotion:Newton’s2ndlaw:

ThePendulummggravitytensionθθmgcosθ-mgsinθTheforceisproportionaltothedistancefrom“equilibrium”freebodydiagramThetangential

forceonthemassis:Forcesonthependulumvisthe“angularvelocity”speed:accelerationintangentialdirection:-thedisplacement

Trythissolution:“Differentialequation”tangentialdirection:Simpleharmonicmotion!

TheperiodisindependentofthemassmTheperiodisindependentoftheamplitudeATheperioddependsonthelengthLTheperiodalsodependsongravity(g)Theperiodofthependulum:Theamplitude(angle)mustbesmallUsingapendulumtomeasuregBallbouncingbackandforth:NOTaSimpleHarmonicOscillatorForcenotprop.toxPeriodjustdependsonthespeedoftheballEnergyconversionintheharmonicoscillatorIdeally,totalmechanicalenergyisconstantpotentialenergykineticenergyKineticenergyofthemass

potentialenergyofthespring

totalenergy

Conserved!potentialenergykineticenergytime

totalenergyCalculatethespeedfromtheen